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Norovirus (NoV) is a significant cause of gastroenteritis globally, and the consumption of oysters is frequently linked to outbreaks. Depuration is the principle means employed to reduce levels of potentially harmful agents or toxins in shellfish. The aim of this thesis is to construct mathematical models which can describe the depuration dynamics of water-borne pathogens, and specifically examine the dynamics of NoV during depuration for a population shellfish. Legislation is currently under consideration within the EU by the Directorate-General for Health and Consumers (DG SANCO) to limit the maximum level of NoV that consumers are exposed to via this route. Therefore it is important to the utility of the thesis that any models constructed should incorporate control measures which could be used to implement minimum NoV levels. Doing so allowed calculation of minimum depuration times that would be required to adhere to the control measures incorporated into the models. In addition to modelling the impact on pathogens during the depuration, we wished to gain some insight into how the variability, and not just the mean levels, of water-borne pathogens can be as important with respect to the length of depuration required to minimise any food safety risks to the consumer. This proved difficult in the absence of any data sets that can be used to calculate variability measures, as little data is currently available to inform these values for NoV. However, our modelling techniques were able to calculate an upper limit on the variability of water-borne pathogens that can be well approximated by lognormal distributions. Finally we construct a model which provided linkage between the depuration process and the accretion of pathogens by shellfish while still within farming waters. This model proposed that the pulses of untreated waste waters released by sewage treatment works due to high levels of rainfall would be transmitted into shellfish whilst filter-feeding.